Frequently in chemistry we must write very large or very small numbers. A convenient method of expressing these types of numbers is by using exponential notation. This notation expresses the numbers as powers of ten or tenths.
The number 1,000 is the result of the following multiplication: 10 x 10 x 10. Written in exponential terms, this is:
This would normally be referred to as "ten to the third power." In chemistry, the base is normally 10.
How can you easily convert a decimal number to an exponential number? Consider the following relationships.
Notice that the exponent is always the number of zeros in the decimal number. Determine the exponential version of the following decimal number.
100,000,000 = 10
Now, what about numbers that are smaller than one? The number 0.01 is the result of multiplying 0.1 x 0.1. Written in exponential terms, this is:
Once again, consider the relationship between decimal and exponential versions of numbers that are smaller than one.
Notice that again, the number of zeros in the decimal number, including the one to the left of the decimal point, determines the exponent (except it is always a negative value). Determine the exponential version of the following decimal number.
0.0000001 = 10
The number 1 written in exponential terms is 100.
Frequently numbers consist of digits other than ones and zeros. The decimal form of numbers like these are frequently expressed in a form of exponential notation, called scientific notation, in which a decimal number between 1 and 10 is followed by a power of ten. For example, the distance from the sun to the earth is 93,000,000 miles. Written in scientific notation, this becomes:
Notice that in this case you cannot simply count zeros. In order to determine the exponent, count the number of places needed to move the decimal point (real or imaginary) until you obtain a number between 1 and 10.
Convert 4,590,000,000 to scientific notation.
This process works for numbers that are either larger or smaller than one.
Convert 0.000000702 to scientific notation.
When the decimal point moves rightward, the exponent always has a negative sign. Try an example of converting decimal numbers to scientific notation.
NOTE: Enter your answer using the format provided in the following examples:
1.25 x 102 would be entered as 1.25e2
1.25 x 10-3 would be entered as 1.25e-3
In some cases you may want to convert a number from scientific notation to decimal notation. If the exponent is positive, move the decimal point rightward the number of places indicated by the exponent. If the exponent is negative, move the decimal point leftward the number of places indicated by the exponent. In both cases it may be necessary to add zeros.
Convert the following scientific notation to decimal notation.
7.33 x 10-4 = 0.000733. This is the result of moving the decimal point four places.
Try another example:
5.48 x 107=
When exponential numbers are multiplied, the exponents are added.
Thus, if 100 is multiplied by 10,000, in exponential terms this would be:
Be careful when working with negative exponents. Consider the following two examples.
If there is a coefficient in front of the exponentials, multiply them, and then multiply the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Let's try the following calculations.
Try another example.
When exponential numbers are divided, the exponents are subtracted.
Let's try dividing 100,000 by 100 using exponential notation.
As with the multiplication of exponentials, be careful when working with negative exponents. Consider the following two examples.
If there is a coefficient in front of the exponentials, divide them, and then divide the exponential component. Finally, put the answer in scientific notation if not already so, and then round off to the appropriate number of significant figures. Let's try the following calculations.